Abstract
In the work we study the completeness conditions for the system of root functions (SRF) of the operator $L_U$ generated by the differential expression $$l(y)=-y"+qy (q\in L_1(0,1))$$ and the integral conditions $$y^{(j-1)}(0)+(l(y),u_j)=0 (u_j\in L_2(0,1), j=1,2)$$ in the space $H=L_2(0,1)$. We show that SRF of the operator $L_U$ is complete in its domain if there exist two rays in the upper half - plane such for all large $\lambda$ on these rays the characteristic determinant is bounded from below by the function $\lambda^{m}e^{-|\mathrm{Im} \lambda|}$, $m\geq\frac{1}{2}$. If the operator $L_U$ is densely defined, then to ensure the completeness of SRF in $H$, it is sufficient to have the mentioned estimate with an arbitrary $m\in \mathbb{R}$. Moreover, we obtain an integral representation for the characteristic determinant as the sine - transform of some function $A$, which is expressed via $u_1$, $u_2$ and the kernel of the transformation operator for the equation $l(y)=\lambda^2y$. Employing this representation, we find explicit (in terms of the functions $u_1,u_2$) completeness conditions for SRF of the operator $L_U$ in $H$ or $D(L_U)$.